Eigenvalue inequalities for Klein-Gordon Operators

TL;DR

This paper establishes universal eigenvalue inequalities for Klein-Gordon operators on bounded domains, extending classical spectral bounds to relativistic quantum operators and including potential energy modifications.

Contribution

It introduces new eigenvalue inequalities for Klein-Gordon operators, including semiclassical bounds and ratios, generalizing known results for fractional Laplacians.

Findings

Proved lower bounds for eigenvalue means involving domain volume.

Derived upper bounds for eigenvalues in terms of averages.

Extended inequalities to operators with external potential energy.

Abstract

We consider the pseudodifferential operators $H_{m,\Omega}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $\Omega$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,\Omega}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial \Omega$. (The operator $H_{0,\Omega}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < \beta_1 < \beta_2 \le >...$ of $H_{m,\Omega}$ and their means $\overline{\beta_k} := \frac{1}{k} \sum_{\ell=1}^k{\beta_\ell}$. Among the inequalities proved are: {\overline{\beta_k}} \ge {\rm cst.} (\frac{k}{|\Omega|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d \ge 2$ and any $k$: \beta_{k+1} \le \frac{d+1}{d-1} \overline{\beta_k}. Furthermore, when $d \ge 2$ and $k \ge 2j$, \frac{\overline{\beta}_{k}}{\overline{\beta}_{j}} \leq \frac{d}{2^{1/d}(d-1)}(\frac{k}{j})^{\frac{1}{d}}. Finally, we present some analogous estimates allowing for an external potential energy field, i.e, $H_{m,\Omega}+ V(\bf x)$, for $V(\bf x)$ in certain function classes.